Forcing Subdivide to create non-zig-zag edges
The problem is that you have a "triangle fan": A bunch of triangles converging on a single vertex (also known as a pole). The subsurf modifier will not work well with that.

You need quads.



Delete the vertex at the center of all the triangles.



Select the inner ring loop and press E to extrude and without moving the mouse press escape

Then scale all of the vertices to 0 (S0)

That will make all of the newly created vertices converge in a single place. And the subsurf should work well.